scholarly journals Improved extrapolation operator design with the WLSQ method

2003 ◽  
Author(s):  
Jan Thorbecke ◽  
Kees Wapenaar
Keyword(s):  
Extrapolation Operator

Improving explicit seismic depth migration with a stabilizing Wiener filter and spatial resampling

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. S111-S120 ◽  
Author(s):  
Gary F. Margrave ◽  
Hugh D. Geiger ◽  
Saleh M. Al-Saleh ◽  
Michael P. Lamoureux
Keyword(s):  
Wiener Filter ◽  
Filter Method ◽  
Complex Conjugate ◽  
Depth Migration ◽  
Time Migration ◽  
Additional Stability ◽  
Band Limited ◽  
Seismic Depth Migration ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator

We present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space-frequency domain. Instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts, one to control phase accuracy and a second to improve stability. The first partial operator is simply a windowed version of the exact operator for a half step. The second partial operator is designed, using the Wiener filter method, as a band-limited, least-squares inverse of the first. The final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second. This resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length. Additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower temporal frequencies. The amount of evanescent filtering is controlled by building two operator tables, one corresponding to significant evanescent filtering and the other to very little evanescent filtering. During the wavefield extrapolation process, most steps are taken with the second table while the first is invoked only for roughly every tenth step. Also, the data are divided into frequency partitions that are optimally resampled in the spatial coordinates to further enhance the performance of the extrapolation operator. Lower frequencies are downsampled to a larger spatial sample size. Testing of the algorithm shows accurate, high-angle impulse responses and run times comparable to the phase shift method of time migration. Images from trial depth migrations of the Marmousi model show very high resolution.

One-way wave-equation migration in log-polar coordinates

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. S59-S67 ◽  
Author(s):  
Diako Hariri Naghadeh ◽  
Mohamad Ali Riahi
Keyword(s):  
Wave Equation ◽  
Polar Coordinates ◽  
Finite Difference Schemes ◽  
Coordinate Systems ◽  
Low Frequencies ◽  
Wave Propagation Direction ◽  
The One ◽  
Wavefield Extrapolation ◽  
High Order Finite Difference ◽  
Extrapolation Operator

We obtained acoustic wave and wavefield extrapolation equations in log-polar coordinates (LPCs) and tried to enhance the imaging. To achieve this goal, it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one-way wave-equation migration (WEM). If we were unable to carry it out, more reflection wave energy would be lost in the migration process. It was concluded that the wavefield extrapolation operator in LPCs at low frequencies has a large wavelike region, and at high frequencies, it can mute the evanescent energy. In these coordinate systems, an extrapolation operator can readily lend itself to high-order finite-difference schemes; therefore, even with the use of inexpensive operators, WEM in LPCs can clearly image varied (horizontal and vertical) events in complex geologic structures using wide-angle and turning waves. In these coordinates, we did not encounter any problems with reflections from opposing dips. Dispersion played important roles not only as a filter operator but also as a gain function. Prestack and poststack migration results were obtained with extrapolation methods in different coordinate systems, and it was concluded that migration in LPCs can image steeply dipping events in a much better way when compared with other methods.

scholarly journals Artifact-free reverse time migration

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. A65-A68 ◽  
Author(s):  
Lele Zhang ◽  
Evert Slob ◽  
Joost van der Neut ◽  
Kees Wapenaar
Keyword(s):  
Initial Estimate ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Multiple Reflections ◽  
Numerical Example ◽  
Time Migration ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator ◽  
New Time

We have derived an improved reverse time migration (RTM) scheme to image the medium without artifacts arising from internal multiple reflections. This is based on a revised implementation of Marchenko redatuming using a new time-truncation operator. Because of the new truncation operator, we can use the time-reversed version of the standard wavefield-extrapolation operator as initial estimate for retrieving the upgoing focusing function. Then, the retrieved upgoing focusing function can be used to directly image the medium by correlating it with the standard wavefield-extrapolation operator. This imaging scheme can be seen as an artifact-free RTM scheme with two terms. The first term gives the conventional RTM image with the wrong amplitude and artifacts due to internal multiple reflections. The second term gives a correction image that can be used to correct the amplitude and remove artifacts in the image generated by the first term. We evaluated the success of the method with a 2D numerical example.

Stability versus accuracy for an explicit wavefield extrapolation operator

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 277-283 ◽  
Author(s):  
Atul Nautiyal ◽  
Samuel H. Gray ◽  
N. D. Whitmore ◽  
John D. Garing
Keyword(s):  
Fourier Transform ◽  
Unconditional Stability ◽  
Space Domain ◽  
Stability Properties ◽  
Depth Migration ◽  
Frequency Space ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator ◽  
Accuracy And Stability

Wavefield extrapolation by recursive (depth‐by‐ depth) application of a convolutional operator in the frequency‐space domain, commonly used for depth migration in a laterally‐varying earth, has interesting accuracy and stability properties. We analyze these properties by investigating the operator and its spatial Fourier transform. In particular, we show that the instability caused by spatially truncating the operator can be remedied unconditionally by applying an appropriately chosen spatial taper. However, unconditional stability is gained only at the expense of accuracy. We also identify frequencies and depth extrapolation step sizes for which the problems of accuracy or stability are the most pronounced.

Comparison of different BEM+Born series modeling schemes for wave propagation in complex geologic structures

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. T71-T82 ◽  
Author(s):  
Geng-Xin Yu ◽  
Li-Yun Fu ◽  
Zhen-Xing Yao
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
The Self ◽  
Dimensionless Frequency ◽  
Interaction Operator ◽  
Born Series ◽  
Extrapolation Operator

Complex geologic structures generally consist of irregular subregions with piecewise constant properties. Two different boundary-element method (BEM) plus Born-series schemes have been formulated for wave-propagation simulation in such piecewise homogeneous media by incorporating a Born series and boundary integral equations. Both schemes decompose the resulting boundary integral equation matrix into two parts: (1) the self-interaction operator, handled with a fully implicit BEM, and (2) the extrapolation operator, approximated by a Born series. The first scheme associates the self-interaction operator with each boundary itself and interprets the extrapolation operator as cross-interactions between different boundaries in a subregion. The second scheme relates the self-interaction operator toeach subregion itself and expresses the extrapolation operator as cross-interactions between different subregions in a whole model. In the second scheme, the matrix dimension of the resulting boundary integral equation is reduced by eliminating the traction field. Both numerical schemes have been validated by dimensionless frequency responses to a semicircular alluvial valley, compared with the full-waveform BEM numerical solution. We then extended these schemes to a complex fault model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments indicate that the [Formula: see text] series modeling schemes significantly improve computational efficiency, especially for high frequencies and with multisource seismic surveys. The tests also confirmed that the second modeling scheme has a faster convergence but may cause more noise in higher-order iterations than the first scheme.

Explicit 3D depth migration with a constrained operator

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. S63-S71 ◽  
Author(s):  
Rune Mittet
Keyword(s):  
Wave Equation ◽  
Outer Part ◽  
Floating Point ◽  
Acoustic Wave Equation ◽  
Impulse Responses ◽  
Lateral Velocity ◽  
Alternative Scheme ◽  
Depth Migration ◽  
Complex Multiplications ◽  
Extrapolation Operator

Numerical anisotropy is one of the main problems in the design of explicit 3D depth-extrapolation operators. This paper introduces a new method based on constraining the number of independent coefficients for the full 3D extrapolation operator. The extrapolation operator is divided into two regions. The coefficients for the inner part of the extrapolation operator are treated the same as the full 3D extrapolation operator. The coefficients for the outer part of the extrapolation operator are constrained to be constant as a function of azimuth for a given radius. This strategy reduces the number of floating-point operations because, for each extrapolation step, the number of complex multiplications are reduced and replaced by complex additions. The numerical workload of this alternative scheme is comparable to the Hale-McClellan scheme. Impulse responses are compared with finite-difference solutions for the two-way acoustic-wave equation. It is demonstrated that the numerical anisotropy for the proposed scheme is negligible and that the constrained-depth-extrapolation operator can be used in media with large lateral velocity contrasts. The design of constrained-depth-extrapolation operators with different maximum propagation angles in inline and crossline directions is explained and exemplified. These types of operators can be used to suppress the propagation of aliased energy in the crossline direction during depth extrapolation while reducing numerical cost.

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